In most inequalities below the hidden constants might depend on \(\rho \)  and \(T\)  as in (REF ), however for simplicity we are not going to always specify them explicitly.
Recall from [1]} that
\({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi -\widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\varphi -\partial _x\widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi -\widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }& \lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{L^2}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{L^2}.\) 

Then, it is sufficient to establish
\({A}+ {B}+ {C}&:={\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\psi -\partial _x\widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial ^2_x\varphi -\partial ^2_x\widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+\Vert |x|^{r}(\varphi -\widetilde{\varphi })\Vert _{L^\infty _T L^2_x} \nonumber \\&\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{H^1}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1}+\Vert |x|^{r}(\varphi _0-\widetilde{\varphi }_0)\Vert _{L^2}.\) 
   
We start controlling \({A}\) .
Similarly to steps
(REF ),
(REF )
and
(REF )
above and using the key observation
\(\psi ^2\overline{\varphi }-(\widetilde{\psi })^2\overline{\widetilde{\varphi }}=\psi ^2\overline{(\varphi -\widetilde{\varphi })}+\overline{\widetilde{\varphi }}(\psi +\widetilde{\psi })(\psi -\widetilde{\psi }),\) 
it is not difficult to deduce
\({A}&\lesssim \Vert \psi _0 - \widetilde{\psi }_0\Vert _{H^1}+ T\Vert \psi ^2\overline{\varphi }-(\widetilde{\psi })^2\overline{\widetilde{\varphi }}\Vert _{L^\infty _T H^1_x} \nonumber \\& \lesssim \Vert \psi _0 - \widetilde{\psi }_0\Vert _{H^1}+ T\Vert \psi ^2\overline{(\varphi -\widetilde{\varphi })}\Vert _{L^\infty _T H^1_x}+ T\Vert \overline{\widetilde{\varphi }}(\psi +\widetilde{\psi })(\psi -\widetilde{\psi })\Vert _{L^\infty _T H^1_x} \nonumber \\& \lesssim \Vert \psi _0 - \widetilde{\psi }_0\Vert _{H^1}+ T\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi -\widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi - \partial _x \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2\Big ) \nonumber \\& \qquad + T\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big ) \nonumber \\& \qquad \qquad \times \Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi -\widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi - \partial _x \widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big ).\) 

Furthermore,
(REF )
and
(REF ) yield to
\({A}\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{L^2}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1}+ T {A},\) 
which allows us to take \(T>0\)  sufficiently small to conclude
\({A}\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{L^2}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1}.\) 
   
We decompose \({B}\)  as in
(REF ),
\({B}& \lesssim {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\psi -\partial _x\widetilde{\psi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x(|\varphi |^2 \varphi -|\widetilde{\varphi }|^2 \widetilde{\varphi })\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }=: {B}_1 + {B}_2\) 

and observe that \({B}_1={A}\) .
On the other hand, applying this time the relations
\(\varphi ^2\partial _x \overline{\varphi }-\widetilde{\varphi }^2\partial _x \overline{\widetilde{\varphi }}& =\varphi ^2\overline{(\partial _x \varphi -\partial _x \widetilde{\varphi })}+\partial _x \overline{\widetilde{\varphi }}(\varphi + \widetilde{\varphi })(\varphi -\widetilde{\varphi })\\|\varphi |^2 \partial _x \varphi -|\widetilde{\varphi }|^2\partial _x \widetilde{\varphi }& =|\varphi |^2(\partial _x \varphi - \partial _x \widetilde{\varphi })+\partial _x \widetilde{\varphi }\, \varphi \,\overline{(\varphi -\widetilde{\varphi })}+\partial _x \widetilde{\varphi }\, \overline{\widetilde{\varphi }} \,(\varphi -\widetilde{\varphi })\) 

and reasoning as in (REF ), give
\({B}_2& \lesssim {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi ^2\partial _x \overline{\varphi }-\widetilde{\varphi }^2\partial _x \overline{\widetilde{\varphi }}\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert |\varphi |^2 \partial _x \varphi -|\widetilde{\varphi }|^2\partial _x \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert } \\& \lesssim \Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x^2 \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )^2 \\& \qquad \times \Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi - \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi - \partial _x \widetilde{\varphi }\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big ).\) 

Thus, taking into account
(REF ) and
(REF )
\({B}\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{L^2}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1}.\) 
   
Finally, repeating the arguments in (REF ) implies that
\({C}&\lesssim \Vert |x|^{r}e^{it \partial _x^2}(\varphi _0-\widetilde{\varphi }_0) \Vert _{L^\infty _T L^2_x}+\Big \Vert |x|^{r} \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi }-\widetilde{\varphi }^2\overline{\widetilde{\psi }})(x,t^{\prime })\, dt^{\prime } \Big \Vert _{L^\infty _T L^2_x}\\&:={C}_1+{C}_2,\) 

and again as in (REF ), it follows
\({C}_1\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{H^1}+\Vert |x|^{r}(\varphi _0-\widetilde{\varphi }_0)\Vert _{L^2}.\) 
The treatment of \({C}_2\)  is parallel to (REF ), after rewriting the difference of nonlinear terms via
(REF ),
\({C}_2& \lesssim \Big (\Vert \varphi \Vert _{L^\infty _T H_x^{1}} \,\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2} \,+\Vert \varphi \Vert _{L^\infty _TH^1_{x}}^2\Big )\Vert  \psi - \widetilde{\psi } \Vert _{L^\infty _T H_x^{1}}\\& \quad +\Big (\Vert \varphi \Vert _{L^\infty _TH^1_{x}}+\Vert \widetilde{\varphi }\Vert _{L^\infty _TH^1_{x}}+\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2}+\Vert |x|^{r}\widetilde{\varphi }\Vert _{L^\infty _T L_x^2}\Big )\Vert \widetilde{\psi } \Vert _{L^\infty _T H_x^{1}} \,\Vert \varphi - \widetilde{\varphi }\Vert _{L^\infty _TH^1_{x}}.\) 

As usual, from here
(REF ), (REF )
and (REF )
lead to
\({C}_2\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{L^2}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1},\) 
so
\({C}\lesssim \Vert \varphi _0-\widetilde{\varphi }_0\Vert _{H^1}+\Vert \psi _0-\widetilde{\psi }_0\Vert _{H^1}+\Vert |x|^{r}(\varphi _0-\widetilde{\varphi }_0)\Vert _{L^2}.\) 
   
In brief, (REF ), (REF ) and (REF ) show (REF ).
